{
  "id": "1405.3446",
  "version": "v2",
  "published": "2014-05-14T10:45:58.000Z",
  "updated": "2014-12-04T10:13:29.000Z",
  "title": "On a diffuse interface model of tumor growth",
  "authors": [
    "Sergio Frigeri",
    "Maurizio Grasselli",
    "Elisabetta Rocca"
  ],
  "comment": "31 pages",
  "categories": [
    "math.AP"
  ],
  "abstract": "We consider a diffuse interface model of tumor growth proposed by A.~Hawkins-Daruud et al. This model consists of the Cahn-Hilliard equation for the tumor cell fraction $\\varphi$ nonlinearly coupled with a reaction-diffusion equation for $\\psi$, which represents the nutrient-rich extracellular water volume fraction. The coupling is expressed through a suitable proliferation function $p(\\varphi)$ multiplied by the differences of the chemical potentials for $\\varphi$ and $\\psi$. The system is equipped with no-flux boundary conditions which entails the conservation of the total mass, that is, the spatial average of $\\varphi+\\psi$. Here we prove the existence of a weak solution to the associated Cauchy problem, provided that the potential $F$ and $p$ satisfy sufficiently general conditions. Then we show that the weak solution is unique and continuously depends on the initial data, provided that $p$ satisfies slightly stronger growth restrictions. Also, we demonstrate the existence of a strong solution and that any weak solution regularizes in finite time. Finally, we prove the existence of the global attractor in a phase space characterized by an a priori bounded energy.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2014-05-14T10:45:58.000Z",
      "comment": "27 pages",
      "journal": null,
      "doi": null
    },
    {
      "version": "v2",
      "updated": "2014-12-04T10:13:29.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "35D30",
      "35K57",
      "35Q92",
      "37L30",
      "92C17"
    ],
    "keywords": [
      "diffuse interface model",
      "tumor growth",
      "weak solution",
      "nutrient-rich extracellular water volume fraction",
      "satisfies slightly stronger growth restrictions"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 31,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2014arXiv1405.3446F"
    }
  }
}